Given positive integers $n$ and $k\in[n]$, how to choose a subset $C\subset\{0,1\}^n$ of size $|C|=k$ to maximize the number of pairs $(c_1,c_2)\in C\times C$ with the supports of $c_1$ and $c_2$ disjoint (equivalently, with $c_1$ and $c_2$ orthogonal)? If $k=1+n+\dotsb+\binom ns$, should one choose $C$ to be the set of all vectors with at most $s$ coordinates equal to $1$?

Some equivalent restatements:

• How to choose a family of $k$ subsets of a fixed $n$-element set to maximize the number of pairs of disjoint subsets?

• How to choose a binary code of length $n$ and size $k$ to maximize the number of pairs of codewords with disjoint supports?

• How to choose a simplicial complex on $n$ vertices with $k$ faces to maximize the number of pairs of disjoint faces?

(For the last reformulations observe that the optimal set $C$ is monotonic, aka "downset").